Introduction Elliptic and Dressed Elliptic String Solutions Features of the Dressed Elliptic Strings Future Extensions Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R × S 2 Georgios Pastras NSCR Demokritos based on arXiv:1907.04817 and arXiv:1907.08508 in collaboration with Dimitrios Katsinis and Ioannis Mitsoulas INPP Annual Meeting, NSCR Demokritos, November 15th 2019 Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R Georgios Pastras
Introduction Elliptic and Dressed Elliptic String Solutions Features of the Dressed Elliptic Strings Future Extensions Project HAPPEN overview The project HAPPEN has started on November the 2nd 2018 and will run until November the 1st 2020. It has a budget of 182k e (50k e + overhead used so far) Three post-docs (G. Pastras, G. Linardopoulos, I. Mitsoulas) are working on the project. Technical progress: - WP1 dressed minimal surfaces: Research has advanced and results are expected to be published within the next few months. - WP1a dressed elliptic strings: Research has been completed and results are published in two papers, Eur.Phys.J. C79 (2019) no.10, 869, JHEP 1909 (2019) 106. - WP2 entanglement in thermal field theory: Research has been completed and results are published in two papers, arXiv:1907.04817, arXiv:1907.08508, both to appear in JHEP . - WP3 Holographic RG flow of entanglement: Research has been completed and results are published in one paper, arXiv:1910.06680. - WP4 Equivalence between entanglement thermodynamics and Einstein equations: This is going to be the main goal of the second year of the program. - WP5 Dissemination: 5 papers, 4 conference presentations and our website happen.inp.demokritos.gr/ Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R Georgios Pastras
Introduction Elliptic and Dressed Elliptic String Solutions Features of the Dressed Elliptic Strings Future Extensions Introduction 1 Elliptic and Dressed Elliptic String Solutions 2 Elliptic Solutions The Dressed Elliptic Solutions The Sine-Gordon Counterparts Features of the Dressed Elliptic Strings 3 The Sine-Gordon Counterparts Closed Strings Stability of the Seeds Spike Interactions Energy and Angular Momentum Future Extensions 4 Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R Georgios Pastras
Introduction Elliptic and Dressed Elliptic String Solutions Features of the Dressed Elliptic Strings Future Extensions Section 1 Introduction Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R Georgios Pastras
Introduction Elliptic and Dressed Elliptic String Solutions Features of the Dressed Elliptic Strings Future Extensions Classical string solutions have shed light to several aspects of the holographic duality. The dispersion relations of the classical strings are related to the anomalous dimensions of operators in the dual CFT. 1 They also serve to develop some intuition on the dynamics of the classical system whose quantum version is the only known mathematically consistent theory of quantum gravity 1 S. Frolov and A. A. Tseytlin, Nucl. Phys. B 668, 77 (2003) [hep-th/0304255] N. Beisert, J. A. Minahan, M. Staudacher and K. Zarembo, JHEP 0309, 010 (2003) [hep-th/0306139] S. Frolov and A. A. Tseytlin, Phys. Lett. B 570, 96 (2003) [hep-th/0306143] Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R Georgios Pastras
Introduction Elliptic and Dressed Elliptic String Solutions Features of the Dressed Elliptic Strings Future Extensions In this work: We focus on strings propagating on R × S 2 , which are Pohlmeyer reducible to the sine-Gordon equation. We invert Pohlmeyer reduction and construct systematically the solutions with elliptic SG counterparts Then we perform a B¨ acklund transformation on the side of the SG equation and find new “dressed” string solutions The new solutions have several interesting features They have interacting spikes. There are interesting interrelations between properties of the strings and their SG counterparts. The dressed solutions reveal the stability properties of their seeds. The energy and angular momenta of the dressed solutions have several qualitative features that could be detectable on the side of the boundary CFT. Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R Georgios Pastras
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions The Dressed Elliptic Solutions Features of the Dressed Elliptic Strings The Sine-Gordon Counterparts Future Extensions Section 2 Elliptic and Dressed Elliptic String Solutions Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R Georgios Pastras
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions The Dressed Elliptic Solutions Features of the Dressed Elliptic Strings The Sine-Gordon Counterparts Future Extensions The Elliptic Strings In a previous work 2 we took advantage of Pohlmeyer reduction to systematically construct the Elliptic String Solutions on R × S 2 . The action for strings propagating on R × S 2 , written as a Polyakov action is ∫︂ d 𝜊 + d 𝜊 − (︂ (︂ X · ⃗ ⃗ X − R 2 )︂)︂ S = T ( 𝜖 + X ) · ( 𝜖 − X ) + 𝜇 . The system is integrable. A signature of the system’s integrability is the fact that it can be reduced to an symmetric space sine-Gordon system , in our case the sine-Gordon equation 3 . Defining the reduced field as the angle between the vectors 𝜖 + ⃗ X and 𝜖 − ⃗ X (︂ )︂ (︂ )︂ := f + ′ f −′ cos 𝜚. 𝜖 + ⃗ 𝜖 − ⃗ X · X It is easy to show that the Pohlmeyer field obeys the sine-Gordon equation 𝜖 + 𝜖 − 𝜚 = 𝜈 2 sin 𝜚. Although we know a lot about the sine-Gordon equation, the Pohlmeyer reduction is a non-local and many-to-one mapping , making its inversion a non-trivial task. However, it can be inverted in the case of elliptic solutions . 2 D. Katsinis, I. Mitsoulas and G.P ., Eur.Phys.J. C78 (2018) no.11, 977, [arXiv:1805.09301 [hep-th]] 3 K. Pohlmeyer, Commun. Math. Phys. 46, 207 (1976) Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R Georgios Pastras
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions The Dressed Elliptic Solutions Features of the Dressed Elliptic Strings The Sine-Gordon Counterparts Future Extensions The Elliptic String Solutions - Periodicity Without posting the details of the derivation, the elliptic string solutions assume the form x 2 − ℘ ( a ) 𝜊 0 + R √︁ √︁ x 3 − ℘ ( a ) 𝜊 1 , t 0 / 1 = R √︄ 𝜊 0 / 1 + 𝜕 2 x 1 − ℘ (︁ )︁ cos 𝜄 0 / 1 = , x 1 − ℘ ( a ) x 1 − ℘ ( a ) 𝜊 1 / 0 − Φ (︂ )︂ √︁ 𝜊 0 / 1 ; a 𝜚 0 / 1 = − sgn ( Im a ) , where the quasi-periodic function Φ is defined as Φ ( 𝜊 ; a ) = − i 2 ln 𝜏 ( 𝜊 + 𝜕 2 + a ) 𝜏 ( 𝜕 2 − a ) 𝜏 ( 𝜊 + 𝜕 2 − a ) 𝜏 ( 𝜕 2 + a ) + i 𝜂 ( a ) 𝜊. Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R Georgios Pastras
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions The Dressed Elliptic Solutions Features of the Dressed Elliptic Strings The Sine-Gordon Counterparts Future Extensions The Elliptic String Solutions - Rigid Rotation - Spikes Writing down the Virasoro constraints in terms of the Pohlmeyer field, yields 2 = R 2 𝜈 2 cos 2 𝜚 2 = R 2 𝜈 2 sin 2 𝜚 ⃒ ⃒ ⃒ ⃒ ⃒ 𝜖 0 ⃗ ⃒ 𝜖 1 ⃗ X 2 , X 2 . ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ Thus, whenever the Pohlmeyer field equals an integer multiple of 2 𝜌 , the derivative 𝜖 1 ⃗ X gets inverted and spikes emerge . The elliptic string retain their shape as they move. All well known solutions on the sphere 4 emerge as special limits of these solutions, including giant magnons, GKP strings, BMN particle, giant hoops. The elliptic strings can be classified to four classes 5 , depending on which worldsheet coordinate the SG counterpart depends whether the SG counterpart is an oscillatory or librating pendulum solution 4 D. E. Berenstein, J. M. Maldacena and H. S. Nastase, JHEP 0204 (2002) 013 [hep-th/0202021] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] D. M. Hofman and J. M. Maldacena, J. Phys. A 39 (2006) 13095 [hep-th/0604135] 5 K. Okamura and R. Suzuki, Phys. Rev. D 75 (2007) 046001 [hep-th/0609026] Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R Georgios Pastras
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions The Dressed Elliptic Solutions Features of the Dressed Elliptic Strings The Sine-Gordon Counterparts Future Extensions The Elliptic String Solutions - Classification static oscillating static rotating counterpart counterpart translationally invariant oscillating translationally invariant rotating counterpart counterpart Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R Georgios Pastras
Recommend
More recommend
